Problem 20 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}$$

Step-by-Step Solution

Answer

The simplified trigonometric expression is 1.

Step 1: Rewrite Using Reciprocal Identities

To simplify the expression \( \frac{\sin x}{\csc x} + \frac{\cos x}{\sec x} \), first express \( \csc x \) and \( \sec x \) in terms of their reciprocal identities. Recall that \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). Substitute these into the expression: \[ \frac{\sin x}{\frac{1}{\sin x}} + \frac{\cos x}{\frac{1}{\cos x}} \].

Step 2: Simplify Fractions

Simplify each fraction in the expression. For \( \frac{\sin x}{\frac{1}{\sin x}} \), multiply numerator and denominator to get \( \sin x \times \sin x = \sin^2 x \). Similarly, for \( \frac{\cos x}{\frac{1}{\cos x}} \), this simplifies to \( \cos x \times \cos x = \cos^2 x \).

Step 3: Use Pythagorean Identity

Now that the expression is \( \sin^2 x + \cos^2 x \), use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Therefore, the entire expression simplifies to \( 1 \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that comes from the famous Pythagorean Theorem. It establishes a relationship between the square of the sine and cosine functions. This identity is especially useful in simplifying trigonometric expressions and proving other identities.

The Pythagorean Identity is stated as: \[\sin^2 x + \cos^2 x = 1\]

This equation holds true for all values of the angle \(x\). Understanding and memorizing this identity can make solving trigonometric expressions much more straightforward. When you encounter expressions that contain \(\sin^2 x\) or \(\cos^2 x\), this identity can help simplify the expression considerably, as you'll see in our example where \(\sin^2 x + \cos^2 x\) was simplified directly to 1.

Reciprocal Identities

Reciprocal Identities are another crucial element in trigonometry. They provide ways to express one trigonometric function in terms of another one. This is particularly helpful when you're faced with an expression that involves functions like cosecant (\(\csc\)), secant (\(\sec\)), or cotangent (\(\cot\)).

Here's a quick rundown of the basic reciprocal identities:

• \(\csc x = \frac{1}{\sin x}\)
• \(\sec x = \frac{1}{\cos x}\)
• \(\cot x = \frac{1}{\tan x}\)

Using these identities can transform a complicated expression into something more familiar and easier to manage. In our exercise, replacing \(\csc x\) with \(\frac{1}{\sin x}\) and \(\sec x\) with \(\frac{1}{\cos x}\) allowed us to simplify the given expression significantly.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to their most basic form. The goal is often to express the equation in a simpler or cleaner format, which can make it easier to understand or solve.

There are several strategies for simplifying trigonometric expressions:

• Use Reciprocal and Pythagorean Identities: These identities can transform the expression into simpler equivalents.
• Combine like terms: Similar terms can often be combined to simplify the structure of the expression.
• Factor whenever possible: Breaking expressions down into multiplicative components can sometimes reveal simpler results.

Applying these techniques systematically can drastically reduce the complexity of a trigonometric expression. In our example, the expression \(\frac{\sin x}{\csc x} + \frac{\cos x}{\sec x}\) was simplified by substituting reciprocal identities and employing the Pythagorean Identity, eventually boiling down to the number 1.